My mother used to sing just a line or two of the chorus, probably when my sister or I complained of feeling sick. I never heard her sing more than that, but she and I never could remember many lines of song lyrics.

I just read the Wiki article and she probably heard the 1935 recording by Mae Questel, which was on the b side of The Good Ship Lollypop.
She also used to sing:

"Miss Polly had a dolly who was sick, sick, sick,
So she called for the doctor to come quick, quick, quick."

Except that I think my mother used to sing: "I had a little dolly", and not "Miss Polly had a dolly" but that was over 50 years ago so my memory might be wrong.

Also, just in case you are wondering what a bisque doll is and what sawdust has to do with it, while some bisque dolls are completely made of bisque or porcelain, most have a bisque or porcelain head, and sometimes hands and feet, but the body is made of cloth and stuffed with sawdust or other material.